thin-layer drying characteristics and modeling of chinese

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 386214, 18 pagesdoi:10.1155/2012/386214

Research ArticleThin-Layer Drying Characteristics and Modeling ofChinese Jujubes

Xiao-Kang Yi,1, 2 Wen-Fu Wu,1 Ya-Qiu Zhang,1Jun-Xing Li,3 and Hua-Ping Luo2

1 College of Biological and Agricultural Engineering, Jilin University, Changchun 130022, China2 College of Mechanical and Electronic Engineering, Tarim University, Alar 843300, China3 Institute of Grain Machinery Research, Jilin Academy of Agricultural Machinery,Changchun 130022, China

Correspondence should be addressed to Wen-Fu Wu, [email protected]

Received 8 December 2011; Accepted 31 January 2012

Academic Editor: Zhijun Zhang

Copyright q 2012 Xiao-Kang Yi et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A mathematical modeling of thin-layer drying of jujubes in a convective dryer was establishedunder controlled conditions of temperature and velocity. The drying process took place both in theaccelerating rate and falling rate period. We observed that higher temperature reduced the dryingtime, indicating higher drying rates of jujubes. The experimental drying data of jujubes were usedto fit ten different thin-layer models, then drying rate constants and coefficients of models testedwere determined by nonlinear regression analysis using the Statistical Computer Program. Asfor all the drying models, the Weibull distribution model was superior and best predicted theexperimental values. Therefore, this model can be used to facilitate dryer design and promoteefficient dryer operation by simulation and optimization of the drying processes. The volumetricshrinkable coefficient of jujubes decreased as the drying air temperature increased.

1. Introduction

Jujube (Zizyphus jujuba Mill.) is a characterized Chinese fruit, whose cultivation area hasreached more than 1.5 million hectares in China. Moreover, a lot of products are processedfrom jujubes, such as candied and drunk jujubes, and jujube tea, juice, and sugar. The vastmajority of jujubes are dried and sold at home and abroad, except that a small proportion arereserved for fresh eating [1]. Jujubes play an important role in human nutrition as sources ofsugar, vitamins, protein, and minerals [2]. For thousands of years, besides been used as food,jujube has been commonly used in traditional Chinese medicine. Jujubes tend to spoil becauseof their high moisture contents, which result in the production losses of 25–30% after harvest

2 Mathematical Problems in Engineering

[3]. Drying is one of the widely used methods for postharvest preservation of agriculturalproducts. It is used to decrease considerable moisture content, reduce microbiological activity,and enable storability of the product under ambient temperatures [4, 5]. Dried food haslonger shelf life in packages and lower transportation, handling, and storage costs [6].

Drying is a complicated process relating to simultaneous heat and mass transfer wherewater is transferred by diffusion from inside the food material to the air-food interface andfrom the interface to the air stream by convection [7, 8]. The amount of energy required todry a product depends on many factors, such as initial moisture, desired final moisture, anddrying air temperature and velocity [9]. Thin layer drying means to dry as one layer of sampleparticles or slices. Thin layer drying equations have been used for drying time predictionfor generalization of drying curves [10]. Various mathematical models describing the dryingcharacteristics of different fruits and vegetables have been proposed to optimize the dryingprocess and design efficient dryers [11, 12]. There are many studies on the drying of fruits andvegetables, such as apricot [13], banana [14], carrot [15], fig [16], golden apples, grape [17],green pepper, stuffed pepper, green bean [18], litchi [19], mushroom, pistachio [20], onion[21], and pumpkin [22]. Several researchers have investigated the drying kinetics of variousagricultural products in order to evaluate the Weibull distribution model for describing thethin-layer drying characteristics [23, 24]. In addition, some researchers proposed the dryingproperties and processing technologies of jujubes [25–28]. The Henderson and Pabis model(see Table 2) has been applied in the drying of jujubes [29]; however, although this modelmay describe the drying curve for the specific experiments conditions, it cannot give a clearand accurate view of the important processes during drying.

The objectives of this study are (1) to determine the effect of air temperature and airvelocity on the drying of jujube and to obtain drying curves; (2) to establish a mathematicalmodel for predicting the thin layer drying characteristics of convection drying of jujubes atdifferent drying air temperature and velocity conditions.

2. Materials and Methods

2.1. Materials

Bioer jujube is one of the main Chinese jujubes varieties, which mainly grow in Shanxi andXinjiang Province. Bioer jujubes used in the experiments were produced in Alar city, Xinjiangprovince and were chosen as drying materials in September, 2010. The appearances of jujubeswere presented in Figure 1. The samples in the same species with full maturity and uniformsize were stored in a refrigerator at 4◦C before starting the experiments. The initial moisturecontent was about 70.12% wet basis (w.b.).

2.2. The Laboratory Dryer

The drying experiments were carried out by a laboratory dryer (BG-II) manufactured atCollege of Biological and Agricultural Engineering, Jilin University, Changchun city, Jilinprovince. A schematic view of the experimental arrangement was shown in Figure 2.

The overall dimensions of the dryer are 2.2 × 0.6 × 1.8 m and it mainly consisted ofa fan, electrical heater, drying chamber, and temperature and humidity control unit. Thefavourable drying air velocity provided by the fan could be changed by the electrical motorwithout level. A 0–15 m/s range anemometer (LUTRON, AM-4201, Taiwan) measured the

Mathematical Problems in Engineering 3

(a) (b)

Figure 1: The appearances of jujubes: (a) fresh jujubes; (b) dry jujubes.

4

5

2

3

1

8 76

Figure 2: Schematic view of the experimental arrangements. 1: Fan; 2: Diffuser; 3: Heater; 4: Bucket; 5:Drying chamber; 6: Humidity control unit; 7: Temperature control unit; 8: Variator.

velocity of air passing through the system. The drying air temperature was automaticallycontrolled by regulating the required voltage to the heaters inside the air channel. Theheater consisted of four groups of resistance wires of 1,000 W, and each group could be usedindependently to control the temperature (30–110◦C, dry bulb temperature) of air and indrying chamber. The dry bulb temperature inside the drying chamber was measured andcontrolled with an accuracy of ±0.1◦C using a Pt 100, 1/10 DIN, thermometer inserted in themiddle position of the inlet cross-section. Temperature-humidity sensor (GALLTEC, TFK80J,Germany) was used to measure the relative humidity with an accuracy of ±3%Rh. Resistancewires were on and off by the control unit based on temperature change to maintain adjustedtemperature at the same level during the experiments. A digital electronic balance (OHAUS,CP3102, USA) in the measurement range of 0–3100 g and an accuracy of 0.01 g was used forthe moisture loss of samples.

2.3. Drying Procedures

Drying experiments were carried out at different drying temperatures of 45, 55, and 65◦Cand different velocities of 0.5, 1.0, and 2.0 m/s. The drying air temperature was automaticallycontrolled at ±1◦C by regulating an electrical heating device and the air velocity wasmeasured by an anemometer at precision of 0.1%.


Table 1: Uncertainties of the parameters during drying of jujubes.

Parameter Unit Comment

Fan inlet temperature ◦C ±0.5Heater outlet temperature ◦C ±0.5Drying chamber inlet temperature ◦C ±0.3Drying chamber outlet temperature ◦C ±0.3Ambient air temperature ◦C ±0.3Inlet of fan with dry and wet thermometers ◦C ±0.5Mass loss values min ±0.1Temperature values min ±0.1Uncertainty in the mass loss measurement g ±0.5Uncertainty in the air velocity measurement m/s ±0.14Uncertainty of the measurement of relative humidity of air RH ±0.1Uncertainty in the measurement of moisture quantity g ±0.001Uncertainty in reading values of table (ρ, cp.) % ±0.1-0.2

The dryer took some time to reach the desired value after starting up. Approximately200 g of samples were put into a stainless-steel mesh bucket of 200 mm diameter, and thenthey were put into the dryer after weighting. In all the experiments, samples were keptthe same thickness and tiled into the layers. The weighing interval of the drying sampleswas 1 h during the drying process. Since the weighing process only took a few seconds,no considerable disturbances were imposed. According to the standards set by GeneralAdministration of Quality Supervision, Inspection and Quarantine (AQSIQ) [30], the dryingprocess was continued until the moisture content of the samples reached below 25% (w.b.).After the drying experiments, the samples were put into an electric constant temperatureblower oven, maintaining at 105 ± 2◦C until their weight remained unchanged. This weightwas used to calculate the moisture content of the samples.

2.4. Experimental Uncertainly

Errors and uncertainties in experiments can arise from instrument selection, condition,calibration, environment, observation, reading, and test planning [31]. In the dryingexperiments of jujubes, the temperatures, velocity of drying air, and weight losses weremeasured with appropriate instruments. During the measurements of the parameters, theuncertainties occurred were presented in Table 1.

2.5. Theoretical Considerations and Mathematical Formulation

Moisture contents of jujubes during thin layer drying experiments were expressed indimensionless form as moisture ratios MR with the following equation [32, 33]:

MR =(M −Me)(M0 −Me)

, (2.1)


Table 2: Mathematical models applied to the moisture ratio values.

Eq. no. Model name Model equation References

(2.1) Lewis MR = exp (−kt) [47](2.2) Page MR = exp (−ktn) [48](2.3) Modified Page MR = α exp [−(ktn)] [49](2.4) Overhults MR = exp [−(kt)n] [50](2.5) Henderson and Pabis MR = α exp (−kt) [51](3.1) Logarithmic MR = α exp (−kt) + c [39](3.2) Two term exponential MR = α exp (−kt) + b exp (–k1t) [52](3.3) Wang and Singh MR = 1 + αt + bt2 [49](3.4) Thompson t = α ln (MR) + b ln (MR)2 [53](3.5) Weibull distribution MR = α − b exp [− (ktn)] [54]

M is the mean jujubes moisture content, M0 is the initial value, and Me is the equilibriummoisture content. The drying rates of jujubes were calculated by using (2.2) [34, 35]

Drying rate =Mt+Δt −Mt

Δt. (2.2)

Mt and Mt+Δt are the moisture content at t and moisture content at t + Δt (kg water/kg drymatter), respectively, t is the drying time (h).

Convection drying of fruits occurs in the falling rate drying period, thus the well-known semiempirical and empirical models could be applied to the drying data. To select asuitable model for describing the drying process of jujubes, drying curves were fitted with10 thin-layer drying moisture ratio models (Table 2). During the analysis of mathematicaldrying models, it was assumed that materials contained the same initial moisture content;there was no heat loss with insulation of dryer walls; material internal temperature gradient,drying air humidity, and heat transfer between materials and volume contraction rate duringdrying were negligible.

The regression analysis was performed with the STATISTICA computer programdeveloped by Statistical Package for Social Science (SPSS) 18. The coefficient of determinationR2 was one of the primary criteria when selecting the best equation to account for variation inthe drying curves of dried samples [36–38]. In addition, the goodness of fit was determinedby various statistical parameters such as reduced chi-square, χ2 mean bias error and rootmean square error (RMSE). For a qualified fit, R2 should be high while χ2 and RMSE are low[39, 40]. These statistical values are calculated as follows:

R2 = 1 −∑n

i=1(MRi,pre − MRi,exp

)2

∑ni=1

(MRi,exp − MRi,premean

)2, (2.3)

χ2 =

∑Ni=1

(MRexp,i − MRpre,i

)2

N − z, (2.4)

RMSE =

√∑n

i=1(MRi,pre − MRi,exp)2

N. (2.5)


MRexp,t is the ith experimental moisture ratio, MRpre,i is the ith predicted moisture ratio, N isthe number of observations, and z is the number of constants in the drying model.

3. Results and Discussion

3.1. Drying Characteristics of Jujubes

The changes in moisture ratios with time for different drying air temperatures are shown inFigure 3. The final moisture content of samples dried under different conditions ranged from28% to 25% (w.b.). The drying rate is higher for higher air temperature. As a result, the timetaken to reach the final moisture content is less, as shown in Figure 3. Therefore, the dryingair temperature has an important effect on the drying of jujubes. The changes of the moistureratios at different air velocities (0.5, 1.0 and 2.0 m/s) under each air temperature (45, 55, and65◦C) were shown in Figure 4. The data revealed that the drying air velocities had little effecton the drying process. Similar results have been reported for plum [41], rosehip [42].

3.2. Drying Rate of Jujubes

The changes in the drying rates with moisture content for different drying air temperaturesand velocities are shown in Figures 5 and 6. It is apparent that the drying process involvedtwo periods, accelerating period and falling period, without a constant-rate drying period.At the beginning of the drying process, the drying rate increases rapidly with decreasingmoisture content and reaches the maximum. Then drying rate decreases continuously withdecreasing moisture content, and the drying operations are seen to occur in the falling rateperiod. It was also noted the predominant direct effect of air temperature on the drying rate,as clearly shown in the figures for all temperatures. Due to the fact that the relative humidityof the drying air at a higher temperature was less compared to that at a lower temperature,the difference in the partial vapor pressure between the radishes and their surroundingswas greater for the higher temperature drying environment [43]. The data revealed that thedrying air velocities had little effect on the drying rate. These results are in agreement withthe previous works [31, 44].

3.3. Mathematical Modelling of Thin-Layer Drying

The mathematical drying models were based on the experimental moisture contents anddry weights. Then continuous data were obtained at different drying air temperaturesand velocities and they were converted into moisture ratios and fitted over drying time.According to the statistical results of the determination coefficient R2, chi-square (χ2), andRMSE, ten thin-layer drying models were compared and shown in Table 3. The data showedthat the highest coefficient (R2), and the lowest chi-square (χ2) and RMSE were obtainedwith the Weibull distribution model. Consequently, it could be concluded that the Weibulldistribution model could sufficiently define the thin layer drying of jujubes. The model couldbe expressed in the following equation:

MR = a − b exp[−(ktn)]. (3.1)


Table

3:St

atis

tica

lres

ults

of10

mod

els

atd

iffer

entd

ryin

gco

ndit

ions

.

Mod

elD

ryin

gai

rve

loci

ty(m

/s)

Dry

ing

air

tem

pera

ture

(◦C)

4555

65R

2χ

2R

MSE

R2

χ2

RM

SER

2χ

2R

MSE

Lew

is0.

50.

9993

830.

0000

360.

0059

820.

9965

560.

0002

070.

0141

720.

9950

020.

0003

030.

0169

651.

00.

9968

380.

0001

920.

0137

570.

9957

630.

0002

530.

0156

440.

9825

340.

0008

540.

0285

242.

00.

9933

840.

0004

310.

0205

940.

9951

830.

0002

940.

0168

850.

9992

490.

0000

420.

0063

56

Page

0.5

0.99

9835

0.00

0010

0.00

3099

0.99

9336

0.00

0041

0.00

6223

0.99

8104

0.00

0121

0.01

0450

1.0

0.99

7864

0.00

0132

0.01

1306

0.99

9782

0.00

0013

0.00

3547

0.99

2789

0.00

0371

0.01

8327

2.0

0.99

8801

0.00

0079

0.00

8765

0.99

9893

0.00

0007

0.00

2516

0.99

9265

0.00

0043

0.00

6291

Mod

ified

Page

0.5

0.99

9835

0.00

0010

0.00

3091

0.99

9433

0.00

0036

0.00

5751

0.99

8495

0.00

0102

0.00

9311

1.0

0.99

9847

0.00

0010

0.00

3024

0.99

9929

0.00

0005

0.00

2025

0.99

6819

0.00

0173

0.01

2172

2.0

0.99

9355

0.00

0044

0.00

6429

0.99

9934

0.00

0004

0.00

1983

0.99

9305

0.00

0043

0.00

6116

Ove

rhul

ts0.

50.

9998

350.

0000

100.

0030

990.

9993

360.

0000

410.

0062

230.

9981

040.

0001

210.

0104

501.

00.

9978

640.

0001

320.

0113

060.

9997

820.

0000

130.

0035

470.

9927

890.

0003

710.

0183

272.

00.

9988

010.

0000

790.

0087

560.

9998

930.

0000

070.

0025

160.

9992

650.

0000

430.

0062

91

Hen

der

son

and

Pabi

s

0.5

0.99

9689

0.00

0019

0.00

4249

0.99

9068

0.00

0058

0.00

7371

0.99

8404

0.00

0102

0.00

9588

1.0

0.99

6848

0.00

0195

0.01

3735

0.99

8067

0.00

0119

0.01

0567

0.98

6812

0.00

0679

0.02

4785

2.0

0.99

5737

0.00

0283

0.01

6531

0.99

8223

0.00

0112

0.01

0256

0.99

9250

0.00

0044

0.00

6355

Log

arit

hmic

0.5

0.99

9739

0.00

0016

0.00

3892

0.99

9441

0.00

0036

0.00

5707

0.99

8589

0.00

0096

0.00

9015

1.0

0.99

9721

0.00

0018

0.00

4083

0.99

9823

0.00

0011

0.00

3196

0.99

3283

0.00

0365

0.01

7688

2.0

0.99

9929

0.00

0005

0.00

2137

0.99

9703

0.00

0019

0.00

4193

0.99

9301

0.00

0043

0.00

6134

Two-

term

expo

nent

ial

0.5

0.99

9689

0.00

0020

0.00

4249

0.99

9068

0.00

0062

0.00

7371

0.99

8404

0.00

0115

0.00

9588

1.0

0.99

6848

0.00

0202

0.01

3735

0.99

8067

0.00

0128

0.01

0567

0.99

7733

0.00

0130

0.01

0276

2.0

0.99

5737

0.00

0293

0.01

6531

0.99

8223

0.00

0120

0.10

1256

0.99

9304

0.00

0045

0.00

6122

Wan

gan

dSi

ngh

0.5

0.99

8691

0.00

0079

0.00

8711

0.99

8516

0.00

0092

0.00

9304

0.99

6917

0.00

0197

0.01

3324

1.0

0.99

5748

0.00

0263

0.01

5952

0.99

9753

0.00

0263

0.00

3775

0.97

0316

0.00

1528

0.03

7185

2.0

0.99

8902

0.00

0073

0.00

8388

0.99

9889

0.00

0007

0.00

2565

0.99

5006

0.00

0030

0.01

6395

Tho

mps

on0.

50.

9993

260.

1578

460.

3895

830.

9995

280.

0455

330.

0268

180.

9989

470.

0389

180.

1871

521.

00.

9995

680.

1385

820.

3661

120.

9997

740.

0193

160.

1344

240.

9946

310.

2176

030.

4437

112.

00.

9997

710.

0712

020.

2623

510.

9993

670.

0575

690.

2323

170.

9995

570.

0250

160.

1517

07

Wei

bull

dis

trib

utio

n

0.5

0.99

9926

0.00

0005

0.00

2076

0.99

9454

0.00

0032

0.00

5643

0.99

8643

0.00

0098

0.00

8839

1.0

0.99

9939

0.00

0004

0.00

1916

0.99

9960

0.00

0003

0.00

1523

0.99

7498

0.00

0144

0.01

0796

2.0

0.99

9943

0.00

0004

0.00

1911

0.99

9935

0.00

0004

0.00

1962

0.99

9306

0.00

0044

0.00

6113


Drying air velocity: 0.5 m/s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio

(a)

Drying air velocity: 1 m/s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio

(b)

45◦C55◦C65◦C


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio

(c)

Figure 3: The experimental moisture ratios at different drying temperatures under each air velocity.

MR is the moisture ratio; k is drying rate constant (h−1); t is time (h); a, n and b isexperimental constants. R2, changed between 0.997498 and 0.999960; χ2, 0.000003–0.000144;RMSE, 0.001523–0.010796.

The Weibull distribution model was analyzed according to the different drying airtemperatures and velocity conditions. The individual constants were obtained (Table 4).Furthermore, the multiple regression analysis was adopted to determine the relationshipbetween drying air temperature, velocity, and the drying constants a, k, n, b based on thedrying experiment data. All possible combinations of different drying variables were testedand included in the regression analysis [45]. The drying constants and coefficients of themodel were as follows:

a = −0.408876 + 0.004101 · T − 0.021508 · v, (3.2)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio

T = 45◦C

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30Drying time (h)

Moi

stur

e ra

tio

T = 55◦C

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30Drying time (h)

Moi

stur

e ra

tio

T = 65◦C

0.5 m/s1 m/s2 m/s

(c)

Figure 4: The experimental moisture ratios at different air velocities under each drying temperature.

b = −1.371752 + 0.003420 · T − 0.021282 · v, (3.3)

k = −0.084196 + 0.002463 · T − 0.001119 · v, (3.4)

n = 1.036553 + 0.000147 · T − 0.008976 · v. (3.5)

These expressions could be used to accurately predict the moisture ratio at any time during adrying process. The consistency of the Weibull distribution model and relationships betweenthe coefficients and drying variables were shown in Table 5. As shown, R2 changed between


0

5

10

15

20

25

0 50 100 150 200 250

Moisture content (%, db)

Drying velocity: 0.5 m/s

Dry

ing

rate

(%·h

−1)

(a)

Drying velocity: 1 m/s

0 50 100 150 200 250


0

2

4

6

8

10

12

14

Dry

ing

rate

(%·h

−1)

(b)

0 50 100 150 200 250


Drying velocity: 2 m/s

0

2

4

6

8

10

12

14

16

45◦C55◦C65◦C

Dry

ing

rate

(%·h

−1)

(c)

Figure 5: The experimental drying rate at different drying temperatures under each air velocity.

0.993345 and 0.999878, χ2 was between 0.000008 and 0.000133, and RMSE was between0.002704 and 0.018925.

In Figure 7, we compared the experimental and predicted moisture ratio at differentair temperatures (45, 55 and 65◦C) under each velocity (0.5, 1.0 and 2.0 m/s). It could beconcluded that the established model was in good agreement with the experimental resultsat all drying conditions. In this picture figure, a higher drying air temperature produced ahigher drying rate and the moisture ratio decreased faster.

To verify the established mathematical drying model, the experimental and predictedvalues of the moisture ratio at some particular drying conditions were compared. Thesevalues were located near a straight line of 45◦, as shown in Figure 8, indicating that the dryingdata were well fitted with the model. Thus, the drying model could be used to well describethe thin-layer drying characteristics of jujubes.


0

2

4

6

8

10

0 50 100 150 200 250Moisture content (%, db)

T = 45◦CD

ryin

g ra

te(%

·h−1)

(a)

T = 55◦C

0 50 100 150 200 250


0

2

4

6

8

10

12

14

Dry

ing

rate

(%·h

−1)

(b)

T = 65◦C

0 50 100 150 200 250Moisture content (%, db)

0

5

10

15

20

25

0.5 m/s1 m/s2 m/s

Dry

ing

rate

(%·h

−1)

(c)

Figure 6: The experimental drying rate at different air velocities under each drying temperature.

3.4. Volume Shrinkage

Jujubes have a porous structure. There are volume shrinkage and deformation during dryingprocess. If ignoring the thermal expansion of materials, the volume shrinkage coefficientequation is expressed as [46]

βV =dV/V

dM, (3.6)

where V is the volume of jujubes and M is the wet-based average moisture content. Assum-ing that βV is constant during the drying process, the above equation can be transformed asfollow:

V = V0e−βV (M0−M). (3.7)


Table 4: Statistical results of Weibull distribution model and its constants and coefficients at different dry-ing conditions.

Drying airtemperature(◦C)

Drying airvelocity(m/s)

A k n b R2 χ2 RMSE

MR = a − b exp [−(ktn)]

450.5 −0.134726 0.032960 1.029610 −1.132454 0.999926 0.000005 0.002076

1.0 −0.434287 0.027195 0.935657 −1.415498 0.999939 0.000004 0.001916

2.0 −0.281029 0.024410 1.011860 −1.279307 0.999943 0.000004 0.001911

550.5 −0.077911 0.045093 1.087933 −1.083897 0.999454 0.000032 0.005643

1.0 −0.197083 0.043597 1.047830 −1.195635 0.999960 0.000003 0.001523

2.0 −0.189766 0.041694 1.063013 −1.190183 0.999935 0.000004 0.001962

650.5 −0.051858 0.080259 1.103842 −1.060497 0.998643 0.000098 0.008839

1.0 −0.451906 0.071394 0.859843 −1.460438 0.997498 0.000144 0.0107962.0 −0.078706 0.081792 1.013627 −1.079853 0.999306 0.000044 0.006113

Table 5: Influences of drying air temperatures and velocities on Weibull distribution model coefficients.

Drying airtemperature (◦C)

Drying airvelocity (m/s) R2 χ2 RMSE

MR = (−0.408876 + 0.004101 · T − 0.021508 · v) − (−1.371752 + 0.003420 · T − 0.021282 · v)·exp{−[(−0.084196 + 0.002463 · T − 0.001119 · v) · t(1.036553+0.000147·T−0.008976·v)]}

450.5 0.998378 0.000098 0.009698

1.0 0.999878 0.000008 0.002704

2.0 0.999001 0.000066 0.007970

550.5 0.999369 0.000039 0.006064

1.0 0.999855 0.000009 0.002891

2.0 0.999737 0.000017 0.003949

650.5 0.998525 0.000094 0.009217

1.0 0.997425 0.000133 0.010952

2.0 0.993345 0.000389 0.018925

The coefficients of volume shrinkage at different drying air temperatures are shown inFigure 9. We can see from the figure that the coefficients, which rang from 0.011 to 0.020,decline with the increase of air temperature under same air velocity, due to the largerchanges of moisture content at higher temperature. The coefficients, measured by least-squaremethod, are shown in Table 6.

Figure 10 shows that the volume changes with time at different drying air temper-atures. The figure shows that the changing trend is basically identical with the change ofmoisture content. It means that with the increase of air temperature, the material shrinkagebecomes more and more obvious during the drying process. Thus, when we choose airtemperature, not only the drying rate but also the morphology and quality of the productsshould be taken into consideration.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio

Drying air velocity: 0.5 m/s

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio


(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50Drying time (h)

Moi

stur

e ra

tio


45◦C experimental

55◦C experimental65◦C experimental45◦C predicted55◦C predicted

55◦C predicted

(c)

Figure 7: The experimental and predicted moisture ratios at different temperatures under each air velocity.

Table 6: Values of volumetric shrinkable coefficient obtained from different temperatures.

Drying air temperature (◦C) βv R2

45 0.020 0.99555 0.014 0.99265 0.011 0.993

4. Conclusions

Thin-layer drying of jujubes were investigated in this study. Ten models selected from theliteratures were referred to illustrate the characteristics of the drying process and establishmathematical drying models of jujubes. Drying process for jujubes involved two periods,accelerating rate and falling rate period, no constant-rate period of drying was observed.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pred

icte

d m

oist

ure

rati

o

Experimental moisture ratio

Drying of jujubes

0.5 m/s 45◦C 0.5 m/s 55◦C1 m/s 45◦C 1 m/s 55◦C2 m/s 45◦C 2 m/s 55◦C

0.5 m/s 65◦C1 m/s 65◦C2 m/s 65◦C

Figure 8: A comparison of experimental and predicted values by the Weibull distribution model at differentdrying conditions.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ln(V

0/V)

(M0 −M) %0 5 10 15 20 25 30 35

45◦C55◦C65◦C

Figure 9: Coefficients of volume shrinkage at different temperature.

After comparing the calculated R2, χ2, and RMSE in each model, Weibull distribution modelshowed the best agreement with the experimental data. Furthermore, the effects of thedrying air temperatures and velocities on the drying constants and coefficients of the Weibulldistribution model were closely examined. We found that this model could be used topredict the moisture ratios of the jujubes during a drying process at any time, particularly atdrying temperatures of 45–65◦C and velocities of 0.5–2.0 m/s. Moreover, our results showedthat the drying air temperature had a bigger effect on drying rate than the velocity. The


0 5 10 15 20 25 30130

140

150

160

170

180

190

200

210

220

V(m

l)

Drying time (h)

45◦C55◦C65◦C

Figure 10: Volume changes with time at different temperatures.

volumetric shrinkable coefficient of jujubes was found to be in the range of 0.011–0.020 atdrying temperatures of 45–65◦C.

Nomenclature

α, b, c, g, p, n: Drying coefficientsk, k1: Drying constantsM: Moisture content at any time, kg water/kg dry matterMe: Equilibrium moisture content, kg water/kg dry matterM0: Initial moisture content, kg water/kg dry matterMt: Moisture content at t, kg water/kg dry matterMt+�t: Moisture content at t+ � t, kg water/kg dry matterMR: Dimensionless moisture ratioMRexp: Experimental dimensionless moisture ratioMRpre: Predicted dimensionless moisture ratioN: Number of observationsχ2: Chi-squareR2: Coefficient of determinationRMSE: Root mean square errorz: Number of drying constantst: Drying time, hT : Temperature, ◦Cv: Velocity, m/sV : Volume, mLV0: Initial volume, mLβv: Volumetric shrinkable coefficient.


Acknowledgment

This work is supported by National Natural Science Foundation of China (no. 10964009).

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